In the name of God

Computer GraphicsModeling1

Today

• Introduction

• Modeling

• Polygon

Issues in Modeling

• Different ways to represent the shape of an object

• How to think about choosing an object representation?

Modeling Definition

• Definition: Modeling is the process of describing an object

• There are different aims for modeling;1. Exact obj description2. Modeling for rendering usage (typically in graphics)3. Other computer graphics applications: “If it looks right it

is right”

Modeling aims1. Exact obj description

– Computer aided design (CAD), Computer Aided Manufacturing (CAM)

– The model is an exact description2. Modeling for rendering usage (typically in

graphics)– The model only exists to produce a picture– It can be an approximation, as long as the visual result is

good3. Other computer graphics applications : “If it looks

right it is right”– Doesn’t work for CAD

Some Questions to Choose a Representation

• How well does it represents the objects of interest?• How easy is it to render (or convert to polygons)?• How compact is it (how cheap to store and transmit)?• How easy is it to create?

– By hand, procedurally, by fitting to measurements, …

• How easy is it to interact with?– Modifying it, animating it

• How easy is it to perform geometric computations?– Distance, intersection, normal vectors, curvature, …

Categorizing Modeling Techniques

• Surface vs. Volume– Sometimes we only care about the surface

• Rendering and geometric computations

– Sometimes we want to know about the volume• Medical data with information attached to the space• Some representations are best thought of defining the space filled,

rather than the surface around the space

• Parametric vs. Implicit– Parametric generates all the points on a surface (volume) by

“plugging in a parameter” eg (sin cos, sin sin, cos)– Implicit models tell you if a point in on (in) the surface

(volume) eg x2 + y2 + z2- 1 = 0

Modeling Techniques

• Polygon meshes– Surface representation, Parametric representation

• Prototype instancing and hierarchical modeling– Surface or Volume, Parametric

• Volume enumeration schemes– Volume, Parametric or Implicit

• Parametric curves and surfaces– Surface, Parametric

• Subdivision curves and surfaces• Procedural models

Polygon Modeling

• Polygons are the dominant force in modeling for real-time graphics

• Why?

Polygons Dominate

• Everything can be turned into polygons (almost everything)

• We know how to render polygons quickly

• Many operations are easy to do with polygons

• Memory and disk space is cheap

• Simplicity

What’s Bad About Polygons?

• What are some disadvantages of polygonal representations?

Polygons Disadvantage

• They are always an approximation to curved surfaces– But can be as good as you want, if you are willing to pay

in size– Normal vectors are approximate– They throw away information– Most real-world surfaces are curved, particularly natural

surfaces

• They can be very unstructured• It is difficult to perform many geometric operations

Properties of Polygon Meshes• Convex/Concave

– Convexity makes many operations easier:• intersection, rendering, volume computations, …

• Closed/Open– Closed if the polygons as a group contain a closed

space• Every edge lies between two faces

• Simple– Faces intersect each other only at edges and vertices– Edges only intersect at vertices

Polygonal Data Structures

• There are many ways to store a model made up of polygons– Choice affects ease of particular operations– dependence on application

• There are typically three components to a polygonal model:– The location of the vertices– The connectivity - which vertices make up which faces– Associated data: normals, texture coordinates, plane

equations, …• Typically associated with either faces or vertices

Polygon Evaluation

• What are the advantages?

• What are the disadvantages?

Evaluation on Polygons

• What are the advantages? – It’s very simple to read, write, transmit, etc.– A common output format from CAD modelers– The format required for OpenGL

• BIG disadvantage: No higher order information– No information about neighbors– No open/closed information

The First Stage

Each face gets split into 4:Each new vertex is degree 6, original vertices are degree 4

Vertex Indirection

• There are reasons not to store the vertices explicitly at each polygon– Wastes memory - each vertex repeated many times– Very messy to find neighboring polygons– Difficult to ensure that polygons meet correctly

• Solution: Indirection– Put all the vertices in a list– Each face stores the list indices of its vertices

• Advantages? Disadvantages?

v0

v2

v1v4

v3

v0 v2 v3 v4v1vertices

0 2 1 0 1 4 1 2 3 1 3 4faces

Indirection Evaluation

• Advantages:– Connectivity information is easier to evaluate because

vertex equality is obvious– Saving in storage:

• Vertex index might be only 2 bytes, and a vertex is probably 12 bytes

• Each vertex gets used at least 3 and generally 4-6 times, but is only stored once

– Normals, texture coordinates, colors etc. can all be stored the same way

• Disadvantages:– Connectivity information is not explicit

neighborhood

• Many algorithms can take advantage of neighbor information– Faces store pointers to their neighbors– Edges may be explicitly stored– Helpful for:

• Building strips and fans for rendering• Collision detection• Mesh decimation (combines faces)• Slicing and chopping• Many other things

– Information can be extracted or explicitly saved/loaded

Normal Vectors

• Normal vectors give information about the true surface shape

• Per-Face normals:– One normal vector for each face, stored as part of face– Flat shading

• Per-Vertex normals:– A normal specified for every vertex (smooth shading)– Can keep an array of normals analogous to array of vertices– Faces store vertex indices and normal indices separately– Allows for normal sharing independent of vertex sharing

A Cube

Vertices:(1,1,1)(-1,1,1)(-1,-1,1)(1,-1,1)(1,1,-1)(-1,1,-1)(-1,-1,-1)(1,-1,-1)

Normals:(1,0,0)(-1,0,0)(0,1,0)(0,-1,0)(0,0,1)(0,0,-1)

Faces ((vert,norm), …):((0,4),(1,4),(2,4),(3,4))((0,0),(3,0),(7,0),(4,0))((0,2),(4,2),(5,2),(1,2))((2,1),(1,1),(5,1),(6,1))((3,3),(2,3),(6,3),(7,3))((7,5),(6,5),(5,5),(4,5))

Computing Normal Vectors

• Many models do not come with normals– Models from laser scans are one example

• Per-Face normals are easy to compute:– Get the vectors along two edges, cross them and normalize

• For per-vertex normals:– Compute per-face normals– Average normals of faces surrounding vertex

• A case where neighbor information is useful

– Question of whether to use area-weighted samples– Can define crease angle to avoid smoothing over edges

• Do not average if angle between faces is greater than crease angle

Storing Other Information

• Colors, Texture coordinates and so on can all be treated like vertices or normals

• Lighting/Shading coefficients may be per-face or per-object, rarely per vertex

• Key idea is sub-structuring:– Faces are sub-structure of objects– Vertices are sub-structure of faces– May have sub-objects– Associate data with appropriate level of substructure

Indexed Lists vs. Pointers• Previous example have faces storing indices of vertices

– Access a face vertex with: mesh.vertices[mesh.faces[i].vertices[j]]

– Lots of address computations– Works with OpenGL’s vertex arrays

• Can store pointers directly– Access a face vertex with: *(mesh.faces[i].vertices[j])– Probably faster because it requires fewer address

computations– Easier to write– Doesn’t work directly with OpenGL– Messy to save/load (pointer arithmetic)– Messy to copy (more pointer arithmetic)

Meshes from Scanning

• Laser scanners sample 3D positions– One method uses triangulation– Another method uses time of flight– Some take images also for use as textures– Famous example: Scanning the David

• Software then takes thousands of points and builds a polygon mesh out of them

• Research topics:– Reduce the number of points in the mesh– Reconstruction and re-sampling!

Scanning

http://www-graphics.stanford.edu/projects/mich/

Level Of Detail

• There is no point in having more than 1 polygon per pixel– Or a few, if anti-aliasing

• Level of detail strategies attempt to balance the resolution of the mesh against the viewing conditions– Must have a way to reduce the complexity of meshes– Must have a way to switch from one mesh to another– An ongoing research topic, made even more important as

laser scanning becomes popular– Also called mesh decimation, multi-resolution modeling

and other things

Level of Detail

http://www.cs.unc.edu/~geom/SUCC_MAP/

Problems with Polygons

• They are inherently an approximation– Things like silhouettes can never be prefect without very

large numbers of polygons, and corresponding expense– Normal vectors are not specified everywhere

• Interaction is a problem– Dragging points around is time consuming– Maintaining things like smoothness is difficult

• Low level information– Eg: Hard to increase, or decrease, the resolution– Hard to extract information like curvature